School of Mathematical & Statistical Sciences Faculty Publications and Presentations

The Aluthge transform of unilateral weighted shifts and the Square Root Problem for finitely atomic measures

Document Type

Article

Publication Date

11-2019

Abstract

In this paper we consider the following Square Root Problem for measures: Given a positive probability Borel measure μ (supported on an interval [a,b]⊆R+), does there exist a positive Borel measure ν such that μ=ν*ν holds? (Here * denotes the multiplicative convolution, properly defined on R+.) This problem is intimately connected to the subnormality of the Aluthge transform of a unilateral weighted shift. We develop a criterion to test whether a measure μ admits a square root, and we provide a concrete solution for the case of a finitely atomic measure having at most five atoms. In addition, we sharpen the statement of a previous result on this topic and extend its applicability via a new technique that uses the standard inequality of real numbers to generate a diagram of a partial order on the support of a probability measure.

Comments

© 2019 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Publication Title

Mathematische Nachrichten

DOI

10.1002/mana.201800140

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